大家帮我看看这几道题,谢谢了..急

kevinzhuzhu

1.
.A television store owner figures that 50% of the customers entering his store will
purchase an ordinary television set, 20% will purchase a high definition television set,
and 30% will just be browsing. If five customers enter the store on a certain day, what is
the probability that two customers purchase an HDTV, one customer purchases a color se
t, and two customers purchase nothing?

2. Suppose that two teams are playing a series of games, each of which is independently
won by team A with probability p and by team B with probability 1-p. The winner of the
series is the first team to win k games.
a) If k=4, what is the probability that a total of 7 games are played? Show that this
probability is maximized when k=1/2.
b) Find the expected number of games that are played when k=2; when k=3; and when
k=4.

3.

Suppose that each coupon obtained is, independent of what has been previously obtained,
equally likely to be any of m different types. Find the expected number of coupons one
needs to obtain in order to have at least one of each type.
Hint: Let X be the number needed. It is useful to represent X by

     m

X=Σ    Xi             , where each Xi is a geometric random variable.

     i=1

(这一题的最后X的表达式我没有用论坛的数学表达式方法输入,但是大家看得懂吧)

4
A coin, having probability p of landing heads, is flipped until head appears for the r th
time. Let N denote the number of flips required. Calculate E[N].
Hint: There is an easy way of doing this – it involves writing N as the sum of r geometric
random variables.

5
Let E and F be mutually exclusive events in the sample space of an experiment.
Suppose that the experiment is repeated until either event E or event F occurs. What
does the sample space of this new “super” experiment look like? Show that the
probability that the event E occurs before the event F is P(E)/{P(E)+P(F)}
Hint: Argue that the probability that the original experiment is performed n times and E
                                                      n-1

occurs on the nth time is P(E)*(1-p) , n=1,2,…, where p = P(E)+P(F). Add these
probabilities to get the desired answer. ( 这里n-1是(1-p)的n-1次方,大家注意.我同样写的不是很清楚.)

6
Suppose that we want to generate a random variable X that is equally likely to be either 0
or 1, and all that we have at our disposal is a biased coin that, when flipped, lands on
heads with some (unknown) probability p. Consider the following procedure:
1. Flip the coin, and let O1, either heads or tails be the result.
2. Flip the coin, and let O2 be the result.
3. If O1 and O2 are the same, return to step 1.
4. If O2 is heads, set X=0 and, otherwise set X=1.
Show that the random variable X generated by this procedure is equally like to be either 0
or 1.

p31415

你先把它翻译后，再弄上去。